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This could be very person dependent. For example I was terrified of spiders as a teenager, but remember dreams where multiple melon sized spiders were around and I only felt mild displeasure, not full blown horror that they would've evoked in the waking state
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Alex StoneFeb 14 '13 at 5:03

1 Answer
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Dreams and nightmares often relate to thoughts, worries, and concerns that a person has during waking hours. Various theories both describe this phenomena and suggest reasons for this (e.g., activation of memories, memory consolidation, discarding information, working through issues). Thus, it seems plausible that someone with a particular phobia would be more likely to have dreams related to that phobia (i.e., fear of heights accompanied by dreams about falling).

Muris et al (2000) discuss some of the research on children looking at fears, worries, and scary dreams. They cite Amaya-Jackson and March (1995) as evidence for links between traumatic events and frightening dreams. They also mention research by Bauer (1976) which suggests that the contents of children's dreams and fears may change with age and that such changes co-occur both in daily fears and in scary dreams (e.g., fewer monsters and more fears about bodily injury/illness with age).

However, in addition to associating a phobia with dream content, you might also want to look at the baseline prevalence of any given type of dream. There is a sleep and dream database you can look at.
In a survey of 2992 participants 48% reported having had a dream of falling (Bulkeley - database - Q9).
Given the relatively high prevalence of dreams of falling, there may be other reasons, perhaps related to the nature of dreaming that induces them. For example, dreams of being unable to move can readily be explained by the paralysed state of body while dreaming. Perhaps a similar argument could be made about falling (e.g., a body sensation while dreaming is interpreted in a way that links with the fear of falling, thereby inducing the nightmare).